Is it possible to overlap two identical Uv Islands in order to create one complete mesh?
Have half of your mesh with UVs done.*

Bake your stuff.
Mirror your model.



Your UVs will be overlapping but not at the time of baking. If you bake while overlapping Blender will get confused and want to put two images (even though they're exactly the same, Blender doesn't know that) at the same place and that's why it comes out weird.

*If you already have your UVs overlapping, applied the Mirror modifier and rather not delete half of the mesh to mirror it again, then you can select one of the halves' UVs, scale them super small and put them in a corner, like this:

Bake your stuff and then ctrl + Z so that the UVs are overlapping again, but your stuff is already baked.

Short answer: Overlapping only causes problems while the baking process happens. Bake only half of your model and then overlap the UVs.